Application of Random Walk Methods to Wave Propagation

Here we present an approach to problems of diffraction that has its roots in a number of well-established theories such as the geometric theory of diffraction, the method of parabolic equations, the theory of Wiener functional integration, and the theory of stochastic processes. We start our analysis of the Helmholtz equation following closely the scheme of the ray method, but instead of approximating the resulting second-order auxiliary equation by a firstorder equation, we study the original auxiliary equation and obtain its exact solution as the mathematical expectation of some functional in the space of Brownian trajectories with the Wiener probabilistic measure. The obtained solution appears to be a direct improvement over the ray method approximation to the exact solution of the Helmholtz equation, and it is shown to admit efficient numerical evaluation.